Field of the Invention
This invention relates to polarized pixelated filter arrays for polarimetric imaging, and more particularly to configurations of the polarized pixelated filter sub-array that reduce sensitivity to misalignment to the imaging detector.
Description of the Related Art
Imaging detectors, such as focal plane arrays, generally include an array of pixels, each pixel including a photo-detector that generates a signal responsive to light generated or reflected by an object. These signals are collected and combined such that a digital image of the object can be created. Pixelated filter arrays positioned in a fixed location over the detector array are widely used in commercial imaging systems to provide hyperspectral or polarimetric capability. For example, digital cameras use fixed-in-place pixelated filter arrays for color (RGB) photography. These filters reduce the amount of light that reaches the imaging pixel (for example, a red filter reflects blue and green light). Polarimeters can be used to analyze the polarization components of light. Some polarimeters use two or more linear polarizers (polarized pixels) that filter at least half of the incoming light and direct the remaining light to a focal plane. As a result, the brightness of the image at the focal plane is substantially reduced (e.g., by about half). Polarimeters can, for example, be used to extract shape information from an object.
U.S. Patent Publication 2014/0063299 to Fest et. al. entitled “Movable Pixelated Filter Array”, which is hereby incorporated by reference, describes polarimetric imaging and introduces a technique for using the data reduction matrix that provides the Stokes vector elements to account for axial and/or lateral misalignment of the pixelated filter array relative to the imaging detector or its conjugate.
Polarimetry requires at least three measurements to analyze the polarization components of light. Typically, the pixelated filter array, and pixelated detector, are divided into groups of four pixels (e.g., a 2×2 sub-array of pixels). Dividing the pixelated detector into larger groups of pixels (e.g., 3×3 or 4×4) is not required and further reduces the spatial resolution of the polarimetric image.
The standard commercially available pixelated filter array is a 2×2 array of linear polarizers having angular values of Θ1=0°, Θ2=45°, Θ3=90° and Θ4=135°, respectively as shown in FIG. 2 of U.S. Patent Publication 2014/0063299. For a perfectly aligned pixelated filter array, it is well known that these angular values are optimum. The polarization states are equally spaced throughout the Poincare sphere, which minimizes the condition number (CN) of the data reduction matrix (DRM). The “condition number” is the L2 condition number of the DRM, which is defined as the ratio of the largest and smallest singular values of the DRM. The singular values are computed using the Single Value Decomposition (SVD) of the DRM. The condition number is directly proportional to the noise in the resulting polarimetric image.
The response for a perfectly aligned polarized pixelated filter array is described by equation 9 of the patent publication in which the detector pixel responses P are the product of the measurement matrix Wp and the Stokes vector elements S. The Stokes vector elements S are found by calculating the DRM as the pseudo inverse of the measurement matrix Wp, and computing the product of the DRM and the detector pixel responses P. There is no closed form solution for the pseudo inverse.
Fest et. al. accounts for axial and/or lateral misalignment of the polarized pixelated filter array relative to the imaging detector or its conjugate by modifying terms in measurement matrix Wp as shown in equation 13 of the patent publication. The output of each detector pixel Pi is the sum of the output of every filter pixel that overlaps it, weighted by the overlap area.